Book spreads in PG(7, 2)

An (n, q, r, s) book is a collection of r-subspaces in PG(n, q) called pages, which cover the whole projective space and intersect in a common s-subspace called the spine such that any point outside the spine is in exactly one page. An (n, q, r, s) book t-spread is a t-spread in PG(n, q) for which there exists an (n, q, r, s) book, such that the points of each page of this book and hence the points of the spine are partitioned by t-subspaces of the t-spread. We commence by showing that an (n, q, r, s) book t-spread exists if and only if the following three conditions hold: (i) (r − s)|(n − s), (ii) (t + 1)|(s + 1), (iii) (t + 1)|(r + 1). In general the number of different kinds of (n, q, r, s) book t-spreads is a tiny proportion of the number of different kinds of t-spreads in PG(n, q). In the rest of this paper we present computer-aided classification results for certain types of (7, 2, 5, 3) book 1-spreads. © 2014 Published by Elsevier B.V.